Let’s assume we want to analyze the energy
in an open channel. Because it is open to the atmosphere we can say that there is no
pressure head. Also, for the sake of simplicity, we will define the energy at any given point
in the channel as relative to the bed slope, and this will eliminate the need to account
for the elevation of the channel as part of the potential energy.
In this situation, the only way to define the potential energy of the channel is to
measure the flow depth. Assuming that the shape of the channel as well as the flowrate
conveyed in the channel both remain constant, the increase in flow depth also increases
the potential energy 1 to 1. If we decrease the flow depth in the channel,
again with the flowrate and the geometry remaining constant, we force the flow into a smaller
area. As we decrease the area, we must compensate by increasing the velocity of flow, which
increases the velocity head in the channel. Alternatively, as we bring the flow depth
up, the area enlarges and, with the Q staying the same, we see a decrease in the velocity.
Because the velocity head is defined with a V-squared term the shape of this relationship
is parabolic as seen here. The summation of the potential and kinetic
head in an open channel gives us what is called
the specific energy profile. If you enjoyed your first term of calculus you may look at
the specific energy curve and ask, I wonder where the minimum point is along
that curve, which I am showing here as yc. Taking the derivative of the energy with respect
to flow depth gives us this relationship. The dA/dy term is actually another way of
expressing the top width of the flow in the channel.
With a little algebra we can define an expression to help us calculate the critical depth, or
the depth at which the specific energy is minimized.
Unfortunately, both the area and top width expressions are in terms of the critical depth,
and they can not be separated out, so the solution to the critical depth is found
iteratively. However, in the case of the rectangular channel,
the base width of the channel is the top width, so we can do a little substitution and a little
simplification to obtain the following expression which will
allow us to calculate the critical depth directly, simply knowing the discharge flowrate and
the base width. Again, this case applies only for rectangular channels.
Going back to the general case – if we look at
the expression of Q squared T divided by g A cubed, we get a dimensionless number.
If we take the square root of this expression, we have defined the Froude number.
In an open channel we define the Froude number as Velocity divided by the square root of
g D, where D is known as the hydraulic depth of the open channel.
The hydraulic depth is defined as the flow area divided by the flow top width.
When the Froude number is equal to 1, then we state that the flow in the channel is “critical”.
When it is less than 1, then we state that it is subcritical,
And when it is greater than 1 we state that it is supercritical.
Visually speaking when a flowrate is classified as supercritical, this corresponds to higher
velocity flowrates which also means that the flow area is small and
the normal depth would be lower than the critical depth.
Alternatively, when the normal depth is greater than the critical depth, then the flow can
be classified as subcritical flow because along with the deeper normal depth comes a
greater flow area, and a slower velocity in the channel, hence the smaller Froude number.
Why on earth do we care about all of this? As it turns out we can engineer systems that
handle supercritical flow as well as subcritical flow without any problem. The problem arises
when we have a channel that for whatever reason, transitions from supercritical to sub critical
flow. This transition is a type of rapidly varied flow and is known as a hydraulic jump.
When we design channels, we pick the channel shape, and the bed slope of the channel. We
want to design our channels so that they are solidly supercritical or subcritical, but
not in between, because when it is in between, the flowrate is not stable. Therefore we must
also calculate the critical slope of our channel to ensure that the slope we have chosen does
not come too close to the critical slope of the open channel.
To calculate the critical slope we solve manning’s equation for the friction slope. Inside the
A and R terms we insert the critical depth, or yc value, to calculate the critical area
and hydraulic radius. All of these tools: Comparing Normal depth
to critical depth, Calculating the Froude Number, and calculating the critical slope,
are methods of assessing whether or not our channel is going to remain within the flow
regime we expected when we sized the channel. As a result, some agencies specify criteria
for one or more of these checks to make sure that we do not design a channel with an unstable